Finite Group Databasebeta

← #11711877

Subgroups — #11711877

Order 768.

347 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85322C211yesmaximal, characteristic
2M16×SL(2,𝔽3)C211yesmaximal, characteristic
3C2×SL(2,𝔽3)⋊C8C211yesmaximal, characteristic
4#73010C212yesmaximal, complemented
5C8×#226C212yesmaximal, complemented
6#17998—31nomaximal, Sylow
7C2×C4×SL(2,𝔽3)C2211yescharacteristic
8#5419C412yes—
9#5391C411yescharacteristic
10C4⋊#226C411yescharacteristic
11C4×#226C2212yes—
12#5122—41nomaximal
13C8×SL(2,𝔽3)C2212yes—
14SL(2,𝔽3)⋊C8C2211yescharacteristic
15SL(2,𝔽3)⋊C8C2211yescharacteristic
16#3104—31no—
17Q8×M16D311yescharacteristic
18#2962—31no—
19#1591—32no—
20#1584—32no—
21#1584—32no—
22C8×Q16—32no—
23#1571—31no—
24#1493—31no—
25C2×Q8⋊C8—31no—
26C8×M16—31no—
27C22×SL(2,𝔽3)C2×C411yescharacteristic
28C2×#226C2×C412yes—
29C2×C3×M16—41no—
30#926—41no—
31C22×C3⋊C8—41no—
32C4×SL(2,𝔽3)C2×C411yescharacteristic
33C4×SL(2,𝔽3)C2311yescharacteristic
34#859C2×C412yes—
35#815—42no—
36C8×Dic3—42no—
37#533—31no—
38C2×C4×Q8C2×D311yescharacteristic
39#484—32no—
40#477—31no—
41#477—31no—
42#462—31no—
43#462—31no—
44#451—32no—
45#446—32no—
46#446—32no—
47#446—62no—
48C8×Q8—32no—
49C8×Q8C2×D312yes—
50C4×Q16—32no—
51C4×Q16—32no—
52#431—31no—
53#424—61no—
54#423—31no—
55C2×C4⋊C8—31no—
56#405—61no—
57C4×M16—31no—
58C2×C4×C8—31no—
59#335—32no—
60#335—32no—
61Q8⋊C8—32no—
62Q8⋊C8—31no—
63Q8⋊C8—32no—
64Q8⋊C8—31no—
65Q8⋊C8—31no—
66Q8⋊C8—31no—
67#322—32no—
68C82—32no—
69C22×C3×C4—41no—
70C2×SL(2,𝔽3)—21no—
71C2×SL(2,𝔽3)C22×C411yescharacteristic
72#226—22no—
73C3×M16—44no—
74C2×C3×C8—42no—
75#217—42no—
76C4⋊Dic3—41no—
77C4⋊Dic3—41no—
78C4×Dic3—42no—
79C2×C3⋊C8—41no—
80C2×C3⋊C8—41no—
81C2×C3⋊C8—44no—
82C22×Q8C4×D311yescharacteristic
83C2×Q16—32no—
84C2×Q16—32no—
85C2×M16—31no—
86C2×M16S411yescharacteristic
87C22×C8—61no—
88C22×C8—61no—
89C4⋊Q8—31no—
90C4⋊Q8—31no—
91#125—61no—
92C22⋊Q8—32no—
93C22⋊Q8—61no—
94C4×Q8—31no—
95C4×Q8—31no—
96C4×Q8—32no—
97C4×Q8—61no—
98C4×Q8C4×D311yescharacteristic
99C4×Q8C22×D311yescharacteristic
100#117—61no—
101C2×C4⋊C4—31no—
102C2×C42—31no—
103#107—32no—
104#107—32no—
105C4⋊C8—32no—
106C4⋊C8—31no—
107C4⋊C8—31no—
108C4⋊C8—31no—
109C4⋊C8—31no—
110C4⋊C8—61no—
111C4⋊C8—61no—
112C4⋊C8—62no—
113C4⋊C8—62no—
114#103—32no—
115#103—32no—
116#103—32no—
117#103—32no—
118C22⋊C8—32no—
119#97—32no—
120#97—62no—
121C4×C8—32no—
122C4×C8—32no—
123C4×C8—31no—
124C4×C8—31no—
125C4×C8—61no—
126C4×C8—62no—
127C23×C3—41no—
128C2×C3×C4—41no—
129C2×C3×C4—42no—
130C2×C3×C4—42no—
131C2×C3×C4—41no—
132C2×Dic3—42no—
133C2×Dic3—42no—
134SL(2,𝔽3)#13111yescharacteristic
135C3×C8—44no—
136C3⋊C8—42no—
137C3⋊C8—42no—
138C2×Q8—21no—
139C2×Q8—31no—
140C2×Q8—32no—
141C2×Q8—61no—
142C2×Q8—61no—
143C2×Q8C2×C4×D311yescharacteristic
144C22×C4—31no—
145C22×C4—31no—
146C22×C4C2×S411yescharacteristic
147Q16—62no—
148Q16—62no—
149Q16—122no—
150M16—62no—
151M16#22614yescomplemented
152C2×C8—32no—
153C2×C8—32no—
154C2×C8—32no—
155C2×C8—61no—
156C2×C8—61no—
157C2×C8—61no—
158C2×C8—61no—
159C2×C8—122no—
160C2×C8—122no—
161C2×C8C2×S412yes—
162C4⋊C4—31no—
163C4⋊C4—31no—
164C4⋊C4—32no—
165C4⋊C4—61no—
166C4⋊C4—62no—
167C4⋊C4—62no—
168C4⋊C4—61no—
169C4⋊C4—61no—
170C4⋊C4—61no—
171C22⋊C4—62no—
172C42—31no—
173C42—31no—
174C42—31no—
175C42—31no—
176C42—62no—
177C22×C3—42no—
178C22×C3—41no—
179C22×C3—81no—
180C22×C3—81no—
181C3×C4—42no—
182C3×C4—41no—
183C3×C4—41no—
184Dic3—82no—
185Dic3—82no—
186C23C4×S411yescharacteristic
187Q8—31no—
188Q8—61no—
189Q8—61no—
190Q8—62no—
191Q8—62no—
192Q8—241no—
193Q8#90711yescharacteristic
194C2×C4—31no—
195C2×C4—31no—
196C2×C4—31no—
197C2×C4—31no—
198C2×C4—61no—
199C2×C4—61no—
200C2×C4—62no—
201C2×C4—62no—
202C2×C4—121no—
203C2×C4C4×S411yescharacteristic
204C2×C4C2×#22612yes—
205C2×C4#98512yes—
206C2×C4C22×S411yescharacteristic
207C8—62no—
208C8—62no—
209C8—62no—
210C8—62no—
211C8—121no—
212C8—121no—
213C8—121no—
214C8C2×#22614yes—
215C2×C3—41no—
216C2×C3—41no—
217C2×C3—41no—
218C2×C3—82no—
219C22—21no—
220C22—21no—
221C22C4×#22612yes—
222C22C2×C4×S411yescharacteristic
223C4—31no—
224C4—31no—
225C4—61no—
226C4—61no—
227C4—61no—
228C4—122no—
229C4—122no—
230C4C4×#22612yes—
231C4C22×#22611yescharacteristic
232C4C2×#98511yescharacteristic
233C3—41noSylow
234C2—22no—
235C2C2×C4×#22611yescharacteristic
236C2#8544211yescharacteristic
237C2#8545311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×S4
commutatorC2×SL(2,𝔽3)C22×C4
FittingQ8×M16D3
socleC22C2×C4×S4
prime radicalC22×SL(2,𝔽3)C2×C4
cosocleC4×SL(2,𝔽3)C23
hypercenterC2×M16S4
hypocenterSL(2,𝔽3)#131
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C2×M16

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #73010C2semidirect
5. C8×#226C2semidirect
151. M16#226semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#853222
M16×SL(2,𝔽3)2
C2×SL(2,𝔽3)⋊C82
#730102
C8×#2262
#179983
#51224